_Posted on September 26, 2024_ ##### Introduction > [!definition] Let $f$ be a real-valued function defined on an open interval $(a, b)$. $f$ is said to be _differentiable_ at $x \in (a,b)$ if and only if the limit $\large \lim_{t\rightarrow x} \frac{f(t)-f(x)}{t - x}$ converges. We associate with $f$ the function $f'$ who's domain is the set of points at which $f$ is differentiable: $\large f'(x)=\lim_{t\rightarrow x}\frac{f(t)-f(x)}{t-x}$ If $f'$ is defined at every point of a set $E \subseteq (a,b)$, we say that $f$ is differentiable on $E$. The quotient $\large \frac{f(t)-f(x)}{t-x}$ is sometimes called the _difference quotient_. ##### Linearity of Differentiation #TODO post on the superposition principle. Change this post as a result. A mapping $F$ is linear if it satisfies [[The Superposition Principle]]. That is, $F$ is both additive and homogeneous: Additivity: $F(x + y) = F(x) + F(y)$ Homogeneity: $F(cx) = cF(x)$ > [!theorem] **Theorem:** _Differentiation is a linear mapping between functions. That is,_ > _(i) Additivity:_ $(f + g)'(x) = f'(x) + g'(x)$ > _(ii) Homogeneity:_ $(c f)'(x) = c f'(x)$ Proof: Both (i) and (ii) follow directly from the linearity of function limits. In detail, we have for additivity:$\large \lim_{t\rightarrow x}\frac{(f +g)(t)-(f + g)(x)}{t-x} = \lim_{t\rightarrow x} \frac{f(t)-f(x)}{t-x} +\frac{g(t)-g(x)}{t-x}$ $\large= \lim_{t\rightarrow x} \frac{f(t)-f(x)}{t-x} +\lim_{t \rightarrow x}\frac{g(t)-g(x)}{t-x}$ and for homogeneity: $\large\lim_{t\rightarrow x} \frac{cf(t)-cf(x)}{t - x} = \lim_{t\rightarrow x} c \frac{f(t)-f(x)}{t-x} = c \lim_{t\rightarrow x} \frac{f(t)-f(x)}{t-x} = cf'(x)$ as desired. $\blacksquare$ Homogeneity can also be demonstrated as a special case of the [[Product Rule for Differentiation|product rule]].